![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(65869, base_ring=CyclotomicField(330))
M = H._module
chi = DirichletCharacter(H, M([130,207]))
![Copy content]()
gp:[g,chi] = znchar(Mod(423, 65869))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("65869.423");
| Modulus: | \(65869\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(65869\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(330\) |
![Copy content]()
sage:chi.multiplicative_order()
![Copy content]()
gp:charorder(g,chi)
![Copy content]()
magma:Order(chi);
|
| Real: | no |
![Copy content]()
sage:chi.multiplicative_order() <= 2
![Copy content]()
gp:charorder(g,chi) <= 2
![Copy content]()
magma:Order(chi) le 2;
|
| Primitive: | yes |
![Copy content]()
sage:chi.is_primitive()
![Copy content]()
gp:#znconreyconductor(g,chi)==1
![Copy content]()
magma:IsPrimitive(chi);
|
| Minimal: | yes |
| Parity: | odd |
![Copy content]()
sage:chi.is_odd()
![Copy content]()
gp:zncharisodd(g,chi)
![Copy content]()
magma:IsOdd(chi);
|
\(\chi_{65869}(52,\cdot)\)
\(\chi_{65869}(117,\cdot)\)
\(\chi_{65869}(423,\cdot)\)
\(\chi_{65869}(1682,\cdot)\)
\(\chi_{65869}(2162,\cdot)\)
\(\chi_{65869}(3117,\cdot)\)
\(\chi_{65869}(3167,\cdot)\)
\(\chi_{65869}(4078,\cdot)\)
\(\chi_{65869}(4550,\cdot)\)
\(\chi_{65869}(4641,\cdot)\)
\(\chi_{65869}(4840,\cdot)\)
\(\chi_{65869}(4983,\cdot)\)
\(\chi_{65869}(6233,\cdot)\)
\(\chi_{65869}(8867,\cdot)\)
\(\chi_{65869}(9341,\cdot)\)
\(\chi_{65869}(10890,\cdot)\)
\(\chi_{65869}(11149,\cdot)\)
\(\chi_{65869}(11261,\cdot)\)
\(\chi_{65869}(11460,\cdot)\)
\(\chi_{65869}(12853,\cdot)\)
\(\chi_{65869}(12876,\cdot)\)
\(\chi_{65869}(13759,\cdot)\)
\(\chi_{65869}(15439,\cdot)\)
\(\chi_{65869}(16450,\cdot)\)
\(\chi_{65869}(16832,\cdot)\)
\(\chi_{65869}(19343,\cdot)\)
\(\chi_{65869}(20350,\cdot)\)
\(\chi_{65869}(21898,\cdot)\)
\(\chi_{65869}(23017,\cdot)\)
\(\chi_{65869}(23182,\cdot)\)
...
![Copy content]()
sage:chi.galois_orbit()
![Copy content]()
gp:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
![Copy content]()
magma:order := Order(chi);
{ chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
\((64877,996)\) → \((e\left(\frac{13}{33}\right),e\left(\frac{69}{110}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 65869 }(423, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{217}{330}\right)\) | \(e\left(\frac{7}{330}\right)\) | \(e\left(\frac{52}{165}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{112}{165}\right)\) | \(e\left(\frac{247}{330}\right)\) | \(e\left(\frac{107}{110}\right)\) | \(e\left(\frac{7}{165}\right)\) | \(e\left(\frac{19}{330}\right)\) | \(e\left(\frac{43}{110}\right)\) |
![Copy content]()
sage:chi(x)
![Copy content]()
gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
![Copy content]()
magma:chi(x)
